Download Modern Algebra I: Homework Sets and Exams (Spring 2009) - Prof. Yongwei Yao and more Exams Algebra in PDF only on Docsity!
MATH 4441 & MATH 6441: MODERN ALGEBRA I
HOMEWORK SETS AND EXAMS
Yongwei Yao
2009 SPRING SEMESTER GEORGIA STATE UNIVERSITY ??
Contents HW Set #01, Hints 1 HW Set #02, Hints 2 HW Set #03, Hints 3 HW Set #04, Hints 4 HW Set #05, Hints 5 Midterm I, Review 6 Midterm I, Hints 7 HW Set #06, Hints 8 HW Set #07, Hints 9 HW Set #08, Hints 10 HW Set #09, Hints 11 Midterm II, Review 12 Midterm II, Hints 13 Midterm II, More Hints 14 HW Set #10, Hints 15 HW Set #11, Hints 16 HW Set #12, Hints 17 Final Exam, Review 18 Final Exam, Hints 19 Extra Credit Set, Hints—not really 20
Note. There are four (4) problems in each homework set. Math 6441 students need to do all 4 problems while Math 4441 students need to do any three (3) problems out the four. If a Math 4441 student submits all 4 problems, then one of the lowest score(s) is dropped. There is a bonus point for a Math 4441 student doing all 4 problems correctly. There are three (3) PDF files for the homework sets and exams, one with the problems only, one with hints, and one with solutions. Links are available below.
PROBLEMS HINTS SOLUTIONS
Notice. The quoted results are from the textbook Basic Abstract Algebra (2nd edition) by P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Cambridge University Press.
Math 4441/6441 (Spring 2009) Homework Set #01 (Due 01/13) Hints
Problem 1.1. Let A = {a, b, c, d}, B = {b, d, e, f, g} and C = {f, g, h} be sets.
(1) Determine A ∩ B and B ∪ C. (2) Determine (A ∩ B) ∪ C and A ∩ (B ∪ C). Are they equal? (3) Determine A \ B and B ∩ C. (Note that A \ B may be also denoted by A − B.) (4) Determine (A \ B) ∩ C and A \ (B ∩ C). Are they equal?
Hint. All should be straightforward. Determine each set by listing its elements explicitly.
Problem 1.2. Let A, B and C be sets.
(1) Show A \ (B ∩ C) = (A \ B) ∪ (A \ C). (2) Show A \ (B ∪ C) = (A \ B) \ C.
Hint. (1). This is similar to the proof of A \ (B ∪ C) = (A \ B) ∩ (A \ C) as shown in class. (2). For every x ∈ A \ (B ∪ C), show x ∈ (A \ B) \ C. Conversely, for every x ∈ (A \ B) \ C, show x ∈ A \ (B ∪ C).
Problem 1.3. For each function fi, determine whether it is injective but not surjective, surjective but not injective, bijective, or neither injective nor surjective. Justify.
(1) f 1 : R → R≥ 0 with f 1 (x) = x^2 for all x ∈ R, in which R≥ 0 = {x ∈ R | x ≥ 0 } = [0, ∞). (2) f 2 : R≥ 0 → R≥ 0 with f 2 (x) = x^2 for all x ∈ R≥ 0. (3) f 3 : R → R with f 3 (x) = 10x^ for all x ∈ R. (4) f 4 : R → R with f 4 (x) = 10x 2 for all x ∈ R. (Note 10x 2 stands for 10(x (^2) ) , not (10x)^2 .)
Hint. You may follow the solution of this example: Let f 5 : R → R with f 5 (x) = |x| for all x ∈ R. Then f 5 is neither injective nor surjective. It is not injective because f 5 (1) = f 5 (−1) while 1 6 = −1. It is not surjective because there is no x ∈ R such that f 5 (x) = − 2
Problem 1.4 (Math 6441). Let A, B and C be sets.
(1) Show A ∪ (B \ C) ⊇ (A ∪ B) \ C. (2) Find explicit examples of A, B and C such that A ∪ (B \ C) ) (A ∪ B) \ C.
Hint. (1). Let x ∈ (A ∪ B) \ C be an (arbitrary) element. Try to show x ∈ A ∪ (B \ C). (2). For example, you may try letting A = { 1 , 2 , 3 }, B = {... } and C = {... }. You may even start with A = { 1 }. (Note that X ) Y means X 6 = Y and X ⊇ Y .)
PROBLEMS HINTS SOLUTIONS
Notice. The quoted results are from the textbook Basic Abstract Algebra (2nd edition) by P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Cambridge University Press.
Math 4441/6441 (Spring 2009) Homework Set #03 (Due 01/27) Hints
Problem 3.1. Determine whether the following statement is true or false with justification: If f : X → Y and g : Y → X are functions such that g ◦ f = IX , then f ◦ g = IY.
Hint. If there ever exists an instance where the statement fails, then the statement is false, in which case one explicit counterexample is enough. Otherwise (i.e., if the statement always holds), you need to prove it rigorously.
Problem 3.2. Let S 3 denote the set of all bijective functions from X = { 1 , 2 , 3 } to itself. Let ϕ ∈ S 3 and ψ ∈ S 3 be defined as follows
ϕ : 1 7 → 2 , 2 7 → 3 , 3 7 → 1 and ψ : 1 7 → 3 , 2 7 → 2 , 3 7 → 1 (1) Determine ϕ ◦ ψ and ψ ◦ ϕ explicitly. Are they equal? (2) Determine ϕ−^1 and ψ−^1 explicitly. (3) Determine ϕ^2 and ϕ^3 explicitly. Is anyone of the two equal to IX? (4) Determine ψ^2 and ψ^3 explicitly. Is anyone of the two equal to IX?
Hint. You may determine/describe a function in S 3 in the format of 1 7 →? , 2 7 →? , 3 7 →?. Recall that ϕ^2 simply denotes ϕ ◦ ϕ while ψ^3 simply denotes ψ ◦ ψ ◦ ψ.
Problem 3.3. Consider integers 24, 60, 26 and 46.
(1) List all (positive and negative) common divisors of 24 and 60. Determine gcd(24, 60). (2) Express gcd(26, 46) as a linear combination of 26 and 46 (with integer coefficients).
Hint. Part (1) is straightforward while part (2) follows from the Euclidean Algorithm.
Problem 3.4 (Math 6441). Let f : X → Y and g : Y → X be functions satisfying g◦f = IX. Show that f ◦ g = IY if and only if f is onto.
Hint. To prove the “only if” implication, assume f ◦ g = IY and show f is onto. To prove the “if” implication, assume f is onto and try to show f ◦ g = IY. In order to show f ◦ g = IY , it suffice to show f ◦ g(y) = y for all y ∈ Y. For every y ∈ Y , what can be said about y in light of f being onto? Now study f ◦ g(y) and try to show f ◦ g(y) = y. It is given that g ◦ f = IX —don’t forget this.
PROBLEMS HINTS SOLUTIONS
Notice. The quoted results are from the textbook Basic Abstract Algebra (2nd edition) by P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Cambridge University Press.
Math 4441/6441 (Spring 2009) Homework Set #04 (Due 02/03) Hints
Problem 4.1. Determine whether each of the statements is true or false with justification.
(1) There exists a sequence of 13 consecutive positive integers less than 100 that contains exactly 2 primes. (2) There are at least 2 primes in every sequence of 7 consecutive positive integers. (3) If m and n are natural numbers, then (mn)! = m! n!.
Hint. (1). To show this is true, it suffices to find a sequence as required. (2). To show this is false, it suffices to find a counterexample. (3). This is all left to you to figure out.
Problem 4.2. Let x = 1 + 4i, y = 2 − 3 i and z = −
3 + 3i. (1) Compute x + y, x − y, xy and x/y. (2) Write z in polar form z = r(cos θ + i sin θ) with 0 ≤ r ∈ R and 0 ≤ θ < 2 π. (3) Compute z^6. Is z^6 a real number? Show your reasoning/computation.
Hint. Straightforward. What is an efficient way to compute z^6 in light of (2)?
Problem 4.3. For all m, n ∈ Z, let m ∗ n = m + n, the sum of m and n. For each of the statements, determine whether it is true or false with justification.
(1) For all a, b ∈ Z, one has a ∗ b ∈ Z. (2) For all a, b, c ∈ Z, one has (a ∗ b) ∗ c = a ∗ (b ∗ c). (3) There exists a (fixed) element e ∈ Z such that e ∗ a = a ∗ e = a for all a ∈ Z. (4) For every a ∈ Z, there exists a′^ ∈ Z such that a ∗ a′^ = a′^ ∗ a = e.
Hint. In (3), it suffices to write down the desired e explicitly if it exists in Z. In (4), it suffices to write down the desired a′^ (depending on a) if it always exists in Z for all a or, otherwise, find an explicit a ∈ Z for which there is no a′^ ∈ Z as claimed. The e in (4) should be the same e as found in (3), provided it exists.
Problem 4.4 (Math 6441). For all m, n ∈ Z, let m ∗ n = mn, the product of m and n. For each of the statements, determine whether it is true or false with justification.
(1) a ∗ b ∈ Z for all a, b ∈ Z. (2) (a ∗ b) ∗ c = a ∗ (b ∗ c) for all a, b, c ∈ Z. (3) There exists a (fixed) element e ∈ Z such that e ∗ a = a ∗ e = a for all a ∈ Z. (4) For every a ∈ Z, there exists a′^ ∈ Z such that a ∗ a′^ = a′^ ∗ a = e.
Hint. In (3), it suffices to write down the desired e explicitly if it exists in Z. In (4), it suffices to write down the desired a′^ (depending on a) if it always exists in Z for all a or, otherwise, find an explicit a ∈ Z for which there is no a′^ ∈ Z as claimed. The e in (4) should be the same e as found in (3), provided it exists.
PROBLEMS HINTS SOLUTIONS
Notice. The quoted results are from the textbook Basic Abstract Algebra (2nd edition) by P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Cambridge University Press.
Math 4441/6441 (Spring 2009) Midterm Exam I (02/17/2009) Review
Functions: Problems 2.1, 2.2, 2.3, 2.4, 3.1, 3.4.
About S 3 : Problem 3.2.
Complex numbers: Problem 4.2.
Definition of groups: Problems 4.3, 4.4, 5.1, 5.2, 5.3, 5.4.
Lecture notes and textbooks: All we have covered.
Note: The above list is not intended to be complete. The problems in
the actual test may vary in difficulty as well as in content. Going over,
understanding, and digesting the problems listed above will definitely help.
However, simply memorizing the solutions of the problems may not help you
very much.
You are strongly encouraged to practice more problems (than the ones
listed above) on your own.
Click here for the actual Midterm Exam I.
Math 4441/6441 (Spring 2009) Midterm Exam I (02/17/2009) Hints
Problem I.1 (5 points). Let x = 5 + 4i, y = −3 + 2i and z = −
3 i. (1) Compute xy and x/y. (2) Write z in polar form z = r(cos θ + i sin θ) with 0 ≤ r ∈ R and 0 ≤ θ < 2 π. (3) Compute z^30 and express it in terms of z^30 = a + bi with a, b ∈ R.
Hint. This is straightforward. What is an efficient way to compute z^30 in light of (2)?
Problem I.2 (5 points). Consider S 3 , which consists of six (6) bijective functions below
f 1 : 1 7 → 1 , 2 7 → 2 , 3 7 → 3; f 2 : 1 7 → 1 , 2 7 → 3 , 3 7 → 2; f 3 : 1 7 → 2 , 2 7 → 1 , 3 7 → 3; f 4 : 1 7 → 2 , 2 7 → 3 , 3 7 → 1; f 5 : 1 7 → 3 , 2 7 → 1 , 3 7 → 2; f 6 : 1 7 → 3 , 2 7 → 2 , 3 7 → 1.
Fill in each of the blanks with (one of) f 1 , f 2 , f 3 , f 4 , f 5 , f 6. (1 point each.)
f 2 ◦ f 3 = , f 3 ◦ f 2 = , f 2 − 1 = , f 5 − 1 = , f 2 − 1 ◦ f 5 − 1 =.
Problem I.3 (5 points). Let f : X → Y and g : Y → Z be functions, in which X, Y and Z are non-empty sets. Prove or disprove each of the following statements.
(1) If f is injective, then g ◦ f is injective. (2) If g is onto, then g ◦ f is onto.
Hint. Provide either a rigorous proof or an explicit counterexample for each statement.
Problem I.4 (5 points). Let X = {x ∈ R | 0. 01 < x < 90 }, an open interval in R. For all x, y ∈ X, define x ∗ y = xy, the usual multiplication. Determine whether each of the statements (1)–(4) is true or false with justification. Then complete (5).
(1) x ∗ y ∈ X for all x, y ∈ X. (2) (x ∗ y) ∗ z = x ∗ (y ∗ z) for all x, y, z ∈ X. (3) There exists a (fixed) element e ∈ X such that e ∗ x = x ∗ e = x for all x ∈ X. (4) For every x ∈ X, there exists x′^ ∈ X such that x′^ ∗ x = x ∗ x′^ = e. (5) (X, ∗) is an abelian group a non-abelian group not a group (choose one)
Hint. Read each statement carefully. Justify your claims over (1)–(4).
Problem I.5 (5 points). Let X = {a + bi | 0 ≤ a ∈ R, b ∈ R} and Y = X \ { 0 }, both in C.
(1) Determine whether (X, +) is a group, where + denotes addition. Show work. (2) Determine whether (Y, · ) is a group, where · denotes multiplication. Show work.
Hint. Key words: closure, associativity, identity and inverse. See Problem I.4.
Extra Credit Problem I.6 (1 point, no partial credit). Let f : X → Y and g : Y → X be functions satisfying g ◦ f = IX. If g is injective, show f ◦ g = IY.
Extra Credit Problem I.7 (1 point, no partial credit). Let (G, ∗) be a group of order 3, i.e., |G| = 3. Show (G, ∗) is abelian.
PROBLEMS HINTS SOLUTIONS
Notice. The quoted results are from the textbook Basic Abstract Algebra (2nd edition) by P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Cambridge University Press.
Math 4441/6441 (Spring 2009) Homework Set #07 (Due 03/10) Hints
Problem 7.1. Consider the group (S 3 , ◦), which consists of six (6) bijective functions below
f 1 : 1 7 → 1 , 2 7 → 2 , 3 7 → 3; f 2 : 1 7 → 1 , 2 7 → 3 , 3 7 → 2; f 3 : 1 7 → 2 , 2 7 → 1 , 3 7 → 3; f 4 : 1 7 → 2 , 2 7 → 3 , 3 7 → 1; f 5 : 1 7 → 3 , 2 7 → 1 , 3 7 → 2; f 6 : 1 7 → 3 , 2 7 → 2 , 3 7 → 1.
Fill in each blank in (1) and (2) with (one of) f 1 , f 2 , f 3 , f 4 , f 5 , f 6. Complete (3) and (4).
(1) f 5 ◦ f 6 = , (f 5 ◦ f 6 )−^1 =. (2) f 5 − 1 = , f 6 − 1 = , f 5 − 1 ◦ f 6 − 1 = , f 6 − 1 ◦ f 5 − 1 =. (3) Does the equation (f 5 ◦ f 6 )−^1 = f 5 − 1 ◦ f 6 − 1 hold? How about (f 5 ◦ f 6 )−^1 = f 6 − 1 ◦ f 5 − 1? (4) Find o(f 1 ), o(f 3 ) and o(f 4 ), the order of the elements.
Hint. This should be straightforward.
Problem 7.2. Let G be a group and let a, b be (fixed) elements of G.
(1) If a−^1 = b−^1 , show a = b. (2) True or false: If a 6 = b, then a−^1 6 = b−^1. Justify your claim. (3) Assume ab = ba. Show (a) ab−^1 = b−^1 a; (b) a−^1 b = ba−^1 ; (c) a−^1 b−^1 = b−^1 a−^1.
Hint. You may want to use the fact that (x−^1 )−^1 = x and (xy)−^1 = y−^1 x−^1 for all x, y ∈ G. (3). From ab = ba, one sees, for example, a−^1 (ab) = a−^1 (ba) among many other valid equations. Keep “playing” with the equations (according to the rules).
Problem 7.3. Let G be an abelian group and let N = {g ∈ G | o(g) < ∞}.
(1) True or false: (ab)n^ = anbn^ for all a, b ∈ G and all n ∈ Z. No need to justify. (2) True or false: e ∈ N. Justify your claim. (3) For all x, y ∈ N , show (a) xy ∈ N ; (b) x−^1 ∈ N.
Hint. Note that an element a ∈ G belongs in N if and only if o(a) < ∞ if and only if there exists an integer n > 0 such that an^ = e. Also note that (a−^1 )m^ = (am)−^1 for all m ∈ Z.
Problem 7.4 (Math 6441). Let G be a group such that (ab)^4 = a^4 b^4 , (ab)^5 = a^5 b^5 and (ab)^6 = a^6 b^6 for all a, b ∈ G. Show G is abelian. (Compare this with Problem 8.4.)
Hint. This is, in spirit, similar to (essentially the same as) an example shown in class.
PROBLEMS HINTS SOLUTIONS
Notice. The quoted results are from the textbook Basic Abstract Algebra (2nd edition) by P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Cambridge University Press.
Math 4441/6441 (Spring 2009) Homework Set #08 (Due 03/17) Hints
Problem 8.1. Let A = C \ { 0 }. It is known (and easy to prove) that (A, · ) is a group under the usual multiplication while (C, +) is a group under the usual addition.
(1) Determine the orders of 1, i, −^12 +
√ 3 2 i^ and 2, considered as elements of (A,^ ·^ ). (2) Determine the orders of 0, 1 and i, if they are considered as elements of (C, +).
Hint. For any group (G, ∗) and x ∈ G, recall that o(x) (if o(x) < ∞) is defined as the least positive integer n such that x︸ ∗ x ∗ · · · ∗︷︷ x︸ n terms
is the identity of (G, ∗).
Problem 8.2. Let G be an abelian group and n a (fixed) integer. Let H = {x ∈ G | xn^ = e}.
(1) Show e ∈ H. (Thus H 6 = ∅.) (2) For all x, y ∈ H, show xy ∈ H. (3) For all x ∈ H, show x−^1 ∈ H.
Hint. For any element a ∈ G, note that a ∈ H if and only if an^ = e.
Problem 8.3. Let G be a group and let g ∈ G (be a fixed element). Define h : G → G by h(x) = gxg−^1 for all x ∈ G. Determine whether each of the following statements is true or false with justification.
(1) h(x) = x if xg = gx. (2) h(x) = x only if xg = gx. (Or, worded differently, xg = gx if h(x) = x.) (3) h(xy) = h(x)h(y) for all x, y ∈ G. (4) h(x−^1 ) = (h(x))−^1 for all x ∈ G.
Hint. This should be straightforward.
Problem 8.4 (Math 6441). Let G be a group. Assume that, for some (fixed) integer n ≥ 1, we have (ab)n^ = anbn, (ab)n+1^ = an+1bn+1^ and (ab)n+2^ = an+2bn+2^ for all a, b ∈ G. Show G is abelian. (Compare this with Problem 7.4.)
Hint. This is a generalized version of Problem 7.4. Study the proof of Problem 7.4.
PROBLEMS HINTS SOLUTIONS
Notice. The quoted results are from the textbook Basic Abstract Algebra (2nd edition) by P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Cambridge University Press.
Math 4441/6441 (Spring 2009) Midterm Exam II (03/31/2009) Review
Explicit computations, orders of elements: Problems 7.1, 7.3, 8.1, 9.1, 9.2.
Basic properties of groups: Problems 7.2, 7.3, 8.2, 8.3,....
Proving a group is abelian: Problems 6.2, 6.3, 7.4, 8.4.
Homomorphisms, kernels, images: Problems 8.3, 9.1, 9.2, 9.3.
Subgroups: Problems 7.3, 8.2, 9.3, 9.4.
Lecture notes and textbooks: All we have covered.
Note: The above list is not intended to be complete. The problems in
the actual test may vary in difficulty as well as in content. Going over,
understanding, and digesting the problems listed above will definitely help.
However, simply memorizing the solutions of the problems may not help you
very much.
You are strongly encouraged to practice more problems (than the ones
listed above) on your own.
Click here for the actual Midterm Exam II.
Math 4441/6441 (Spring 2009) Midterm Exam II (03/31/2009) Hints
Problem II.1. Consider the group (S 3 , ◦) under composition, which consists of the following
f 1 : 1 7 → 1 , 2 7 → 2 , 3 7 → 3; f 2 : 1 7 → 1 , 2 7 → 3 , 3 7 → 2; f 3 : 1 7 → 2 , 2 7 → 1 , 3 7 → 3; f 4 : 1 7 → 2 , 2 7 → 3 , 3 7 → 1; f 5 : 1 7 → 3 , 2 7 → 1 , 3 7 → 2; f 6 : 1 7 → 3 , 2 7 → 2 , 3 7 → 1.
Define h : (Z, +) → (S 3 , ◦) by h(m) = f 5 m for all m ∈ Z, which is a group homomorphism. Fill in each blank with an integer or fi with 1 ≤ i ≤ 6. Complete (4).
(1) o(f 5 ) = , o(f 6 ) =. (3) h(−8) = , h(11) =.
(2) h(−6) = , h(9) =. (4) List 10 elements in Ker(h).
Hint. Which is the identity of (S 3 , ◦)? Note that amq+r^ = (am)qar.
Problem II.2. Let h : G → G′^ be a group homomorphism (with G and G′^ being groups). If h is surjective and G is abelian, show G′^ is abelian.
Hint. Let x, y ∈ G′^ be arbitrary elements. What can be said about x and y in light of h being surjective? Show xy = yx by making use of the (given) assumptions.
Problem II.3. Let h : G → G′^ be a group homomorphism and K < G′. (As always, denote the identity elements of G and G′^ by e and e′^ respectively.) Define H = {x ∈ G | h(x) ∈ K}.
(1) Show H < G. (2) Show Ker(h) ⊆ H.
Hint. (1). Note the defining property of H: an element x ∈ G is in H if and only if h(x) ∈ K. (2). You only need to show Ker(h) is a subset of H.
Problem II.4. Let G be an abelian group and N be a (fixed) subgroup of G. Denote H = {a^28 | a ∈ N } and F = {a^4 | a ∈ N }.
(1) Show H < G. (2) Show H ⊆ F.
Hint. (1). Note the defining property of H: x ∈ H if and only if x = a^28 for some a ∈ N. (2). You only need to show H is a subset of F. What is the defining property of F?
Problem II.5. Let G be a group. Define h : G → G by h(x) = x^2 for all x ∈ G.
(1) If h is a group homomorphism, show G is abelian. (2) If G is abelian, show h is a group homomorphism.
Hint. (1). What is the meaning of h being a homomorphism in light of the definition of h?
Extra Credit Problem II.6 (1 point, no partial credit). Prove or disprove: If h : G → G′ is a group homomorphism and G is abelian, then G′^ is abelian.
Hint. Work on this. Either give a rigorous proof or give an explicit counterexample.
Extra Credit Problem II.7 (1 point, no partial credit). Prove or disprove: If h : G → G′ is a group homomorphism and x ∈ G such that o(h(x)) < ∞, then o(x) < ∞.
Hint. Work on this. Either give a rigorous proof or give an explicit counterexample.
PROBLEMS HINTS SOLUTIONS
Notice. The quoted results are from the textbook Basic Abstract Algebra (2nd edition) by P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Cambridge University Press.
Math 4441/6441 (Spring 2009) Homework Set #10 (Due 04/07) Hints
Problem 10.1. Consider the group (S 3 , ◦) under composition, which consists of the following
f 1 : 1 7 → 1 , 2 7 → 2 , 3 7 → 3; f 2 : 1 7 → 1 , 2 7 → 3 , 3 7 → 2; f 3 : 1 7 → 2 , 2 7 → 1 , 3 7 → 3; f 4 : 1 7 → 2 , 2 7 → 3 , 3 7 → 1; f 5 : 1 7 → 3 , 2 7 → 1 , 3 7 → 2; f 6 : 1 7 → 3 , 2 7 → 2 , 3 7 → 1.
Find as many (distinct) subgroups of S 3 as possible. You will receive 1 point per correct subgroup and −1 point per incorrect choice. (It is possible to get more than 5 points.)
Hint. You may describe the subgroups in the format of H 1 = {f 3 , f 6 }, H 2 = {f 2 , f 4 , f 6 },
... , H 5 = {f?,... , f?}, etcetera, in which H 1 and H 2 could be wrong answers. Note that any element a in any group G generates a cyclic subgroup [a].
Problem 10.2. Consider the group (S 3 , ◦) as in Problem 10.1. Let H = {f 1 , f 2 } < S 3.
(1) For each fi ∈ S 3 with 1 ≤ i ≤ 6, determine the left coset fiH explicitly. (2) How many distinct left cosets of H are there in the group (S 3 , ◦)? (3) Write S 3 as a disjoint union of the distinct left cosets found above.
Hint. This is straightforward. Just carry out your computations carefully. To answer (3), just write S 3 = {f?, f?} ∪ · · · ∪ {f?, f?}, in which each {f?, f?} should be a left coset of H.
Problem 10.3. Let G be a group of order 30, i.e., |G| = 30.
(1) If x ∈ G satisfies x^12 = e, x^14 6 = e, x^15 6 = e, determine o(x). (2) If y ∈ G satisfies y^12 = e, y^15 6 = e and y^20 = e, determine o(y).
Hint. Use Lagrange’s Theorem. Also, what does an^ = e tell you about o(a)?
Problem 10.4 (Math 6441). Let G be a group of order 4, i.e., |G| = 4.
(1) If there exists a ∈ G such that o(a) = 4, show G = {e, a, a^2 , a^3 }. (2) If no element of G has order 4, use Lagrange’s Theorem to show x^2 = e for all x ∈ G.
Now, in light of (1) and (2), show G is abelian. (Compare with Problem E-2.)
Hint. (1). If o(a) = 4, then what can be said about [a]? In particular, how many and what distinct elements does [a] consist of? Can you show G is abelian in this case? (2). For every x ∈ G, what can be said about o(x) in light of Lagrange’s Theorem and the assumption o(x) 6 = 4? Can you show x^2 = e? Can you show G is abelian in this case?
PROBLEMS HINTS SOLUTIONS
Notice. The quoted results are from the textbook Basic Abstract Algebra (2nd edition) by P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Cambridge University Press.
Math 4441/6441 (Spring 2009) Homework Set #11 (Due 04/14) Hints
Problem 11.1. Consider the group S 3 under composition, which consists of the following
f 1 : 1 7 → 1 , 2 7 → 2 , 3 7 → 3; f 2 : 1 7 → 1 , 2 7 → 3 , 3 7 → 2; f 3 : 1 7 → 2 , 2 7 → 1 , 3 7 → 3; f 4 : 1 7 → 2 , 2 7 → 3 , 3 7 → 1; f 5 : 1 7 → 3 , 2 7 → 1 , 3 7 → 2; f 6 : 1 7 → 3 , 2 7 → 2 , 3 7 → 1.
Let H = {f 1 , f 2 }, which is a subgroup of S 3.
(1) Determine whether H is a normal subgroup of (S 3 , ◦). Show your reasoning. (2) Find the least i with 1 ≤ i ≤ 6 such that fiH 6 = Hfi.
Hint. This is straightforward. Here fiH and Hfi are short for fi ◦H and H ◦fi respectively.
Problem 11.2. Let G be a group and Hi < G for i = 1, 2.
(1) Show H 1 ∩ H 2 < G. (2) Further assume Hi G for i = 1, 2. Show H 1 ∩ H 2 G.
Hint. (1). Just use the subgroup criterion we learned in class. (2). From (1), we see that H 1 ∩ H 2 < G. In order to show H 1 ∩ H 2 G, let h ∈ H 1 ∩ H 2 and x ∈ G (be arbitrary elements). It suffices to show xhx−^1 ∈ H 1 ∩ H 2.
Problem 11.3. Consider the group (Z, +). Let H = { 4 n | n ∈ Z}, which is a subgroup of (Z, +). In other words, H = {... , − 8 , − 4 , 0 , 4 , 8 ,... } consisting of all multiples of 4.
(1) Determine the (left) cosets 0 + H, 1 + H, 3 + H and 4 + H explicitly. (2) True or false: (−3) + H = 1 + H. No justification is needed. (3) True or false: (−2) + H = 2 + H. No justification is needed. (4) How many distinct (left) cosets of H are there inside the group (Z, +)? (5) Is Z a disjoint union of the distinct (left) cosets? No justification is needed.
Hint. This is straightforward. Just carry out the computations carefully.
Problem 11.4 (Math 6441). Let G be a group and N G. For every (fixed) x ∈ G, show xN = N x.
Hint. To prove xN = N x, we need to show xN ⊆ N x and xN ⊇ N x. For example, you may show xN ⊆ N x as follows: Let y ∈ xN. Then y = xn for some n ∈ N (by the definition of xN ). Therefore we see y = xn = · · · = (xnx−^1 )x. Can you show/see y ∈ N x by now? why? (Use the assumption that N is a normal subgroup of G.) Now use a similar argument to show xN ⊇ N x: Let z ∈ N x and try to show z ∈ xN.
PROBLEMS HINTS SOLUTIONS
Notice. The quoted results are from the textbook Basic Abstract Algebra (2nd edition) by P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Cambridge University Press.
Math 4441/6441 (Spring 2009) Final Exam (04/28/2009) Review
Explicit groups Sn and Zn: Problems 7.1, 8.1, II.1, 10.1, 10.2, 11.1, 12.2.
Proving a group is abelian: Problems 6.2, 6.3, 7.4, 8.4, II.2, II.5, 10.4.
Subgroups: Problems 7.3, 8.2, 9.3, 9.4, II.3, II.4, 11.2.
Normal subgroups: Problems 11.2, 11.4, 12.3, 12.4.
Lagrange’s theorem, orders: Problems 10.3, 10.4, 12.1.
Quotient groups: Problems 12.2, 12.3, 12.4.
Others: Homomorphisms, cyclic subgroups, cyclic groups, etc.
Lecture notes and textbooks: All we have covered.
Note: The above list is not intended to be complete. The problems in
the actual test may vary in difficulty as well as in content. Going over,
understanding, and digesting the problems listed above will definitely help.
However, simply memorizing the solutions of the problems may not help you
very much.
You are strongly encouraged to practice more problems (than the ones
listed above) on your own.
The final exam is cumulative.
Click here for the actual Final Exam.
Math 4441/6441 (Spring 2009) Final Exam (04/28/2009) Hints
Problem III.1. Consider (Z 12 , +), which is a quotient group of (Z, +) modulo its normal subgroup N = 12Z = { 12 n | n ∈ Z}. Thus Z 12 = {0 + N, 1 + N,... , 11 + N }.
(1) Find o(9 + N ) and o(10 + N ). (2) Determine [9 + N ] and [10 + N ] by listing their distinct elements explicitly.
Hint. Note that both 9 + N and 10 + N are elements in Z 12 , whose operation is +. For an element a ∈ Z 12 , you may/should write a︸ + a +︷︷ · · · + a︸
n terms
as na.
Problem III.2. Let G be a group of order 40, i.e., |G| = 40.
(1) Is it ever possible to have an element x ∈ G such that x^28 = e and x^12 6 = e? If so, what are the possible values of o(x)? Show your reasoning. (2) Is it ever possible to have an element y ∈ G such that y^30 = e and y^15 6 = e? If so, what are the possible values of o(y)? Show your reasoning.
Problem III.3. Let G be a group, H < G and N < G such that N ⊆ Z(G). (1) Show HN < G. (2) Show N G.
Hint. Recall Z(G) = {z ∈ G | zg = gz for all g ∈ G} and HN = {hn | h ∈ H, n ∈ N }. An element x ∈ G is in HN if and only if x = hn for some h ∈ H and n ∈ N.
Problem III.4. Let G, H and N be as in Problem III.3 above. (Thus HN < G.) Further assume H = [a] for some (fixed element) a ∈ G. Show HN is abelian.
Hint. Let x, y ∈ HN. Then it suffices to show xy = yx. Recall that [a] = {an^ | n ∈ Z}.
Problem III.5. Let h : G → G′^ be a group homomorphism (with G and G′^ groups) and M < G. Denote N = {h(m) | m ∈ M }. (Denote the identities of G and G′^ by e and e′.)
(1) Show N < G′. (2) Assume h is onto and M G. Show N G′.
Hint. An element x ∈ G′^ is in N if and only if x = h(m) for some m ∈ M.
Problem III.6. Let G be an abelian group, N = {x ∈ G | x^6 = e} and a ∈ G (a fixed element) such that o(a) = 4. It is known that N G by Problem 8.2 and Problem 12.3. (1) Find o(a^2 ) and o(a^3 ). Show work. (2) For aN ∈ G/N , determine o(aN ).
Hint. You may find o(a^2 ) and o(a^3 ) from definition. To determine o(aN ) (from definition), find the smallest positive integer n such that (aN )n^ is the identity of G/N. Note that the identity of G/N is eN and, for u, v ∈ G, uN = vN if and only if uv−^1 ∈ N.
Extra Credit Problem III.7 (1 point, no partial credit). True or false: If G is a group and Hi < G for i = 1, 2, then H 1 ∪ H 2 < G. Give a proof or a counterexample.
Extra Credit Problem III.8 (1 point, no partial credit). Let G be a group, N G and a, b ∈ G such that ab ∈ N. Show ba ∈ N.
PROBLEMS HINTS SOLUTIONS
Notice. The quoted results are from the textbook Basic Abstract Algebra (2nd edition) by P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Cambridge University Press.
